Welcome to The Estimation of Frequency

**Description: **Periodic phenomena occur naturally. Often the periods are obvious. For example, hourly temperature data are expected to have approximate periods of length 24 and 24*365.25 hours approximately. These periods are only approximate, as there is always going to be some variation that is impossible to model deterministically. Often the periodicity or frequency is unknown, and its value crucial to the understanding of the phenomenon. It therefore needs to be estimated from data.

The estimation of frequency from data had occupied scientists since the eighteenth century. The periodogram was introduced by Schuster in 1898 as a means of estimating a `hidden' periodicity, but other techniques were available by then. In particular, the Buys-Ballot method was introduced in 1847. Indeed, Gauss developed an FFT-like technique in about 1805. The statistical theory of the properties of estimators appears to have been evaluated first by Whittle in 1952, who noted that the relevant Cramer-Rao lower bound for frequency was of order T^{-3}, rather than the usual T^{-1}, where T is the sample size. Walker (1971) and Hannan (1975) followed up with rigorous results and proofs for the case where the additive noise has very general properties.

The publication of the Cooley and Tukey FFT algorithm in 1965 has undoubtedly been responsible for the enormous interest in `frequency domain' techniques - those based on the Fourier transforms of data rather than the data themselves. Large numbers of articles have appeared, mainly in the signal processing literature, but also in statistical journals.

In this course, we shall examine many different frequency estimation techniques and establish the asymptotic properties of the estimators. All of the probabilistic and estimation theory needed will be introduced to establish the (strong) consistency and central limit theorems of the estimators. We shall also consider the computational aspects of the algorithms, and use Matlab code to implement them.

**Prerequisites**: A basic knowledge of mathematics as obtained through undergraduate engineering studies.

**Organizer: **Professor Mads Græsbøll Christensen, email: mgc@create.aau.dk

**Lecturer: **Professor Barry Quinn

**ECTS:** 3

**Time: **4. - 8. September 2017

**Place: **Rendsburggade 14, 9000 Aalborg

4 September room 4.531

5 September room 5.125

6, 7 and 8 September room 5.441

**Zip code: **

**City: **Aalborg

**Number of seats: **25

**Deadline:** 15. August 2017

**Important information concerning PhD courses** We have over some time experienced problems with no-show for both project and general courses. It has now reached a point where we are forced to take action. Therefore, the Doctoral School has decided to introduce a no-show fee of DKK 5,000 for each course where the student does not show up. Cancellations are accepted no later than 2 weeks before start of the course. Registered illness is of course an acceptable reason for not showing up on those days. Furthermore, all courses open for registration approximately three months before start. This can hopefully also provide new students a chance to register for courses during the year. We look forward to your registrations.

- Teacher: Mads Græsbøll Christensen
- Teacher: Barry Quinn
- Teacher: Barry Gerard Quinn